%------------------------------------------------------------------------------
% File     : SEU592^2 : TPTP v9.0.0. Released v3.7.0.
% Domain   : Set Theory
% Problem  : Preliminary Notions - Ops on Sets - Unions and Intersections
% Version  : Especial > Reduced > Especial.
% English  : (! A:i.! B:i.subset (binintersect A B) A)

% Refs     : [Bro08] Brown (2008), Email to G. Sutcliffe
% Source   : [Bro08]
% Names    : ZFC094l [Bro08]

% Status   : Theorem
% Rating   : 0.00 v8.2.0, 0.15 v8.1.0, 0.09 v7.5.0, 0.00 v6.0.0, 0.14 v5.5.0, 0.33 v5.4.0, 0.40 v5.1.0, 0.20 v4.1.0, 0.00 v4.0.1, 0.33 v4.0.0, 0.00 v3.7.0
% Syntax   : Number of formulae    :   10 (   3 unt;   6 typ;   3 def)
%            Number of atoms       :   14 (   3 equ;   0 cnn)
%            Maximal formula atoms :    3 (   3 avg)
%            Number of connectives :   25 (   0   ~;   0   |;   0   &;  20   @)
%                                         (   0 <=>;   5  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   3 avg)
%            Number of types       :    2 (   0 usr)
%            Number of type conns  :    9 (   9   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    7 (   6 usr;   2 con; 0-2 aty)
%            Number of variables   :   11 (   3   ^;   8   !;   0   ?;  11   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : http://mathgate.info/detsetitem.php?id=292
%------------------------------------------------------------------------------
thf(in_type,type,
    in: $i > $i > $o ).

thf(dsetconstr_type,type,
    dsetconstr: $i > ( $i > $o ) > $i ).

thf(subset_type,type,
    subset: $i > $i > $o ).

thf(subsetI2_type,type,
    subsetI2: $o ).

thf(subsetI2,definition,
    ( subsetI2
    = ( ! [A: $i,B: $i] :
          ( ! [Xx: $i] :
              ( ( in @ Xx @ A )
             => ( in @ Xx @ B ) )
         => ( subset @ A @ B ) ) ) ) ).

thf(binintersect_type,type,
    binintersect: $i > $i > $i ).

thf(binintersect,definition,
    ( binintersect
    = ( ^ [A: $i,B: $i] :
          ( dsetconstr @ A
          @ ^ [Xx: $i] : ( in @ Xx @ B ) ) ) ) ).

thf(binintersectEL_type,type,
    binintersectEL: $o ).

thf(binintersectEL,definition,
    ( binintersectEL
    = ( ! [A: $i,B: $i,Xx: $i] :
          ( ( in @ Xx @ ( binintersect @ A @ B ) )
         => ( in @ Xx @ A ) ) ) ) ).

thf(binintersectLsub,conjecture,
    ( subsetI2
   => ( binintersectEL
     => ! [A: $i,B: $i] : ( subset @ ( binintersect @ A @ B ) @ A ) ) ) ).

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